A theorem of the transversal theory for matroids of finite character

Let M = (S, I) be a matroid of finite character on the infinite set S. Let A = 〈A1:i ∈ I〉 be any system of subsets of S each having finite rank and let B = 〈B1: j ∈ J〉 be a finite system of sets of arbitrary rank. Necessary and sufficient conditions are given for the system A ⋃ B to have an independent system of distinct representatives.